The Hecke operator \(T(m)\)  can be written as a sum of two operators[1]}, [2]}, the first of which, \(V(m)\)  gives a
new series by replacing each \(q^n\)  in \(f\)  by \(q^{mn}\)
\(\left(V(m)f\right) (z) = \sum _{n=1}^\infty a(n) q^{mn} = \sum _{n=1}^\infty a(n) e^{2\pi i mnz} = f(mz) \) 
